A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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$Y_n = \sup_{k \geq n} E(X_k | F_n)$ is a martingale if $X_n$ is $L^1$ bounded non-negative submartingale

Let $X_n$ be a $L^1$ bounded non-negative submartingale. Let $Y_n = \sup_{k \geq n} E(X_k | F_n)$. Show that (1) $Y_n$ is a martingale (2) $X_n \leq Y_n$ for all $n$ a.s. (3) $\sup \|X_n\|_1 = ...
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13 views

Black Scholes Solution

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
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11 views

What is the nonlinear estimator for Gaussian Random variable?

I know that the best estimator is $g(x)=E\{Y|X=x\}$ and the conditional density for jointly Gaussian random variables is known to be Gaussian with mean and variance given by ...
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9 views

A stochastic process is generated as follows: we assign the value 1 to a head and the value 0 to a tail. Start at n=0, Compute Rxx(0,0) and Rxx(2,3)

I am kind of confused here, since autocorrelation describes the correlation between values of the process at different times, but for the first case, it is at the same time. I got that ...
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29 views

Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that (1) for all $\lambda >0$ $$ P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ (2) $\|\sup X_n\|_1 ...
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9 views

Proof about a homogeneous Poisson process

We know that a homogeneous Poisson process is a process with a constant intensity $\lambda$. That is, for any time interval $[t, t+\Delta t]$, $P\left \{ k \;\text{events in}\; [t, t+\Delta t] \right ...
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28 views

one inequality involving two stochastic processes

I am having trouble proving one inequality involving two stochastic processes. The problem seems simple but I just cannot handle it. Any help would be appreciated. $S_t$ and $C_t$ are two positive ...
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25 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
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52 views

Independence of a Stochastic Process at Distinct Time

Suppose $X_t$ is a stochastic process of $t$ on $[0,\infty)$ with almost surely continuous sample path. I have modified my question to the following one, per Math1000's comment below: Is the ...
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11 views

Memory less property of a Markov chain- Validation methods

Are there any tests to check the memory less property of a discrete time homogeneous Markov chain? I found a chi squared test to verify the time homogeneity of a Markov chain constructed from a set of ...
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29 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
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7 views

Markov Chains: Expected Return Time (Stochastic Process)

I am given a matrix with space {0,1,2,3,4}. I already calculated the invariable probability vector. However, the question asks to give the expected number of steps: -given Xo=0 to go back to state ...
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1answer
38 views

Deriving the Doob Meyer decomposition of a Sub Martingale using Ito's

Given the standard brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$ and defining the sub-m.g.: $$X_t =B^6_t+2t$$ I would like to derive its Doob-Meyer decomposition: [Sub-m.g.]= [increasing ...
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28 views

Size of families: Birth death immigration

The context of this problem is as follows. Starting from a population size of zero, immigrants arrive according to a homogeneous Poisson process with rate $\theta$. Once they arrive, immigrants start ...
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17 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...
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16 views

Stochastic Process

I would like to know if anyone here could help me with this exercise. Y(t) = X(t +d) - X(t), where X(t) is a Gaussian Stochastic process. (A) Calculate the mean and covariance of Y(t) (B) Calculate ...
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25 views

Entry time and hitting time

Hi I have a question about entry time and hitting time. Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_{t})_{t \in[0,\infty)}$ be a $\mathbb{R}$-valued stochastic process on $(\Omega, ...
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1answer
12 views

Definition of mth order stationarity

in the definition of the weak GARCH processes they use the terminology of the 4th-order stationarity of the process $(X_t)$. I know the definition of 2n-order stationarity, but I'm not exactly sure, ...
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18 views

Convergence in distribution of stochastic equation solutions

I'm studying from Kurtz's book "Markov Processes Characterization and convergence" and I have a question about the convergence of processes in $\mathbb{Z}^d$ that are solution of some equation. (see ...
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41 views

Show martingale space is a Banach space

Let $\mathcal{H}^1 = \{M \in \mathcal{M}, E[sup_{t\geq 0} |M_t|] < \infty\}$, where $\mathcal{M}$ is the space of right continuous with left limits martingales. Show that $\mathcal{H}^1$ is ...
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19 views

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$ i) Find the regions where the joint pdf of $(Z,W)$ is positive. ii) Find the ...
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integral with respect to the point measure [on hold]

We have integral $$\int_0^tf(t-u)dX(u)$$ where $X(u)$ is random point process( or at least renewal process). Also it is known that $f(t)\sim t^{-\alpha},$ $0<\alpha<1$ as $t\rightarrow \infty$. ...
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15 views

stochastic process involving cdf of a process [on hold]

I would like to know if anyone here could help me out with this exercise. Here it goes: A stochastic process is created from Yn = c(n)Xn, where Xn is a stochastic process with mean equals to zero, ...
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25 views

Almost sure convergence of stochastic integral

Consider a probability space $(\Omega, \mathcal{F}, P)$ equipped with a Brownian motion $W$. Let $(\xi_n)_{n=1}^\infty$ be a sequence of adapted $\mathcal{F}(t)$-progressively measurable processes ...
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10 views

Aggregation Urn Distribution

I am trying to identify this distribution in terms of the number of balls, $n$, urns, $m$, and iterations, $i$. Before the first iteration each ball is independent. The first iteration consists of ...
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1answer
37 views

Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.

Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then $$ \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ ...
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1answer
21 views

Expected Value of the exponential of a stochastic integral

What is the expected value of the following process: $$ e^{\int_0^t B_u\, du} $$ Thanks.
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30 views

Brownian motion: first-hitting-time with double barrier [closed]

Let $(B_t)_t$ be a standard ($B_0=0$) Brownian motion , and $$ T_{a,b} = \inf\{t>0 : B_t \not\in(a,b)\} $$ where $a<0<b$. What is the expected first-passage time $\mathbf{E}[T_{a,b}]$?
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Find the value of the real $\alpha$ for which exp($2B(t) - \alpha t$) is a martingale.

I tried to answer using the three conditions to be a martingale (measurability, integrability, and martingality), validating the integrability condition, which is $$ E |e^{2B(t) - \alpha t} | < ...
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1answer
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What's the meaning of the state space with locally compact topological space?

I have encountered a statement in one paper describing the continuous-time controlled Markov chain with space state which is locally compact topological space. What does this mean? In my previous ...
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0answers
19 views

Deterministic integrals involving a Brownian motion [duplicate]

I am trying to work out the following two integrals involving a standard Brownian motion started at $W_0 = 0$. The first expression is bewildering me a bit, since it seems like somehow the Itô ...
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1answer
26 views

Pricing a riskless asset in the Black & Scholes market

Consider a Black&Scholes Market where a risky asset evolves according to: $$\frac{dS_t}{S_t}=\mu dt+\sigma dB_t$$ $$S_o=s$$ Riskless asset is associated with risk free rate r. I want to represent ...
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40 views

Expected value and Variance of a stochastic time integral of a deterministic variable (Standard Brownian motion)

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, define: $$E(e^{\int_0^tudB_u})=?$$ $$ Var(e^{\int_0^tudB_u})=?$$ Sidenote to be edited later: Here is my try, I'm not capable to ...
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14 views

Need a little bit of guidance with stochastic processes

Let $X(t) = \begin{bmatrix} cos(t) + N(t)\\ sin(t) + S(t)\\ \end{bmatrix} $ (where $N(t)$ is a gaussian process and S(t) is a Poisson's process). Let ...
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68 views

Expected value of a brownian motion times the deterministic integral of a brownian motion

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, $E (B_t \int_0^tB_s^3ds)$ = ? I try to turn the expected value into a double integral by rewriting the $B_t$ term as 1) $E(\int_0^t ...
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25 views

Solve the stochastic differential equation

I have to solve the following SDE: $$dX_t=X_t dt+2W_tdW_t$$ Let $Y_t=X_t e^{-t}$. By Ito formula we have: $$dY_t=-X_te^{-t}dt+e^{-t}(X_t dt+2W_tdW_t)=2e^{-t}W_tdW_t$$ Thus ...
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44 views

Expected value of an exponential of a gaussian random variable

$$E (Y_t)=E(e^{X_t}) = E(e^{N(X_0e^{at};\frac{b^2}{2a}(e^{2at}-1)}) =\text{ ?}$$ Knowning that $$X_t \sim N\left[X_0e^{at};\frac{b^2}{2a}(e^{2at}-1)\right]$$ $$X_t= aX_t \, dt+b \, dB_t$$ The ...
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1answer
36 views

Proof that there exists a non-negative eigenvector corresponding to eigenvalue 1 of stochastic matrix

Let $P \in [0,1]^{n \times n}$ be a [irreducible or reducible] stochastic matrix where its rows sum to 1 i.e. $$ \forall i \in \{ 1 , \dots n \} \quad \sum_{j=1}^{n} P_{ij} = 1 $$ It is easy to show ...
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18 views

Definition of Simple Predictable Process

I am reading Protter's book "Stochastic Integration and Differential Equations". He (page 51) defines $H$ to be a simple predictable processes if it has a representation ...
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About moments in a quantile processes

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F: $$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$ In this case, $F^{-1}$ is the (generalized) inverse of ...
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How to efficiently simulate successes of several trials if probabilities are inhomogeneous

If I'm doing a simulation with $n$ trials, each with probability $p$, a quick way to select the successful trials is to choose a binomially distributed random number. Then randomly choose that many ...
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40 views

Distribution of $(\sup_{0\leq s\leq t} W_s -W_t)$

I am interest in the law of the $(\sup_{0\leq s\leq t} W_s -W_t)$ where $W$ is a standard brownian motion. I know that $M_t:=\sup_{0\leq s\leq t} W_s \overset{\mathcal L}{=} |W_t |$ so its density ...
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1answer
42 views

Proving that Doob's martingale is a martingale

I'm working on my first ever proof that a stochastic process is a martingale, and I'm a bit confused. Is there a "standard machine" for these proofs? To be more specific, I am trying to show that if ...
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17 views

Inverse Bessel Process

Is there any reference on this process? For example, analytical derivations for the hitting times, density, etc? Im studying local martingales and am interested in the density of hitting times for ...
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21 views

Distribution of points in a homogeneous PPP

PPP holds some important properties. However, my question is whether the positions of the points in a homogeneous PPP are independent? Equivalently, are the points in a homogeneous PPP distributed ...
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26 views

Question about Lebesgue Dominated Convergence Theorem involving a Markov Time / Stopping Time

I am trying to understand the proof of the following lemma: Let $W$ be an arbitrary random variable satisfying $\mathbb{E}[|W|] < \infty$, and let $T$ be a Markov time (or stopping time) for which ...
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46 views

Couple/Compare two stochastic processes and prove an intuitive proposition

Consider a stochastic process (denoted $X$) $X_0, X_1, X_2, \ldots$ (not necessarily a Markov Chain) over state space $\{0, 1, \cdots, n \}$. The transition probabilities are $$P(X_{i+1} = 1 \mid ...
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70 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
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21 views

Stability of non-autonomous stochastic differential equation

I'm looking for a good reference or insight to under what conditions can I prove stability (or instability) for the following general n-dimensional non-autonomous stochastic differential equation: ...
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1answer
30 views

Product topology and uniform topology on C[0,T]

Is the product topology on $\mathbb{R}^{[0,T]}$ restricted to $C[0,T]$ (T finite) the same as the topology induced by the uniform norm on $C[0,T]$? I am curious because I saw a claim on wiki saying ...